Proof : Let ℬ be a finite dimensional C*-algebra. Let a be an element of J⁢(ℬ), the Jacobson radical of ℬ.

J⁢(ℬ) is an ideal of ℬ, so a*⁢a∈J⁢(ℬ).

The Jacobson radical of a finite dimensional algebra is nilpotent, therefore there exists n∈ℕ such that (a*⁢a)n=0. Then, by the C* condition and the fact that a*⁢a is selfadjoint,

0=∥(a*⁢a)2n∥=∥a*⁢a∥2n=∥a∥2n+1

so a=0 and J⁢(ℬ) is trivial. □

We now prove the above claim.

Proof of the claim: It is easy to see that
{[0A00]:A∈M⁢a⁢tn×n⁢(ℂ)} is the only maximal ideal of 𝒜. Therefore the Jacobson radical of 𝒜 is not trivial.

By the theorem we conclude that there is no involution * that makes 𝒜 into a C*-algebra.□

Remark - It could happen that there were no involutions in 𝒜 and so the above claim would be uninteresting. That’s not the case here. For example, one can see that [ai,j]⟶[a¯2⁢n+1-j,2⁢n+1-i] defines an involution in 𝒜 (this is just the taken over the other diagonal of the matrix).

Title

example of Banach algebra which is not a C*-algebra for any involution